Representation Theorems

Abstract

One of the everyday miracles of mathematics is that some objects have two (or more) representations that somehow manage to reveal different features of the object. Our constant use of Taylor expansions and Fourier series tends to blunt their surprise, but in many cases the effectiveness of the right representation can be magical. Consider the representation for min(s, t) that we found from Parceval’s theorem for the wavelet basis and which formed the cornerstone of our construction of Brownian motion; or, on a more modest level, consider the representation of a convex function as the upper envelope of its tangents and the automatic proof it provides for Jensen’s inequality.